What is the length of the contour $γ(t)=5e^{it}$ for $t$ in the interval $[0,2\pi]$?

Question

Let $C$ be the contour $γ(t)=5e^{it}$ for $t$ in the interval $[0,2\pi]$. What is the length of $C$?

Would the length of $C$ be $5$ or $10$? I think $r=5$ so I am not sure whether that would be the length or $2r$, the diameter. Can someone clear this up for me please?

Answer 1

Since $C$ is a circle of radius $5$, its length is the circumference, $2\pi * 5 = 10\pi$.

Answer 2

well the curve is the circle with radius $5$ around the origin. so what is the length of that circle?

Answer 3

Recall that the length of a smooth curve $\gamma:[a,b]\to\mathbb{C}$ is $$\ell(\gamma)=\int_a^b |z'(t)|\mathrm{d}t,$$ where $z(t)$ is any parametrization of the curve.

It might be good practice to try applying this definition to the curve you supplied, even though there are other ways to compute its length (like recognizing that it is the circle of radius 5 centered at the origin, as others have pointed out).